Sharp bounds for Hardy-type operators on mixed radial-angular spaces

In this paper, by using the rotation method, we calculate that the sharp bound for $n$-dimensional Hardy operator $\mathcal{H}$ on mixed radial-angular spaces. Furthermore, we also obtain the sharp bound for $n$-dimensional fractional Hardy operator $\mathcal{H}_\beta$ from $L^p_{|x|}L_{\theta}^{\bar{p}}({\Bbb R}^n)$ to $L^q_{|x|}L_{\theta}^{\bar{q}}({\Bbb R}^n)$, where $0<\beta<n$, $1<p,q,\bar{p},\bar{q}<\infty$ and $1/p-1/q=\beta/n$. By using duality, the corresponding results for the dual operators $\mathcal{H}^*$ and $\mathcal{H}^*_\beta$ are also established. In addition, the sharp weak-type estimate for $\mathcal{H}$ is also considered.